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A175101 The number of bases b for which the odd squarefree semiprime A046388(n) is a Fermat pseudoprime. 1
2, 2, 2, 2, 2, 2, 2, 2, 14, 2, 2, 14, 2, 34, 2, 2, 2, 2, 2, 2, 2, 34, 2, 2, 14, 2, 2, 2, 2, 2, 14, 2, 2, 2, 14, 2, 2, 2, 34, 2, 14, 2, 2, 34, 2, 2, 34, 14, 2, 2, 2, 2, 2, 34, 2, 14, 2, 2, 2, 2, 2, 2, 2, 2, 98, 2, 14, 2, 14, 2, 2, 2, 2, 34, 2, 2, 2, 2, 2, 34, 2, 14, 2, 98, 2, 34, 2, 2, 142, 14, 2, 14, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A number x is a Fermat pseudoprime for base b if b^(x-1) = 1 (mod x).

Comment from Karsten Meyer: (Start) Each term pq of the sequence A046388 is at least a Fermat pseudoprime to the two bases which have the property that |l*p - m*q| = 2 and b is the number between l*p and m*q. There are no more bases of this form below pq.

There may exist other bases smaller than pq, but just two bases have the property that they are direct neighbors of a multiple of p and a multiple of q. For example, 39=3*13 is a Fermat pseudoprime to the bases 14 and 25 because 14 is the number between 13 and 3*5 and 25 is the number between 3*8 and 2*13.

91=7*13 is a Fermat pseudoprime to the bases 27 and 64 because 27 is the number between 2*13 and 4*7 and 64 is the number between 9*7 and 5*13. For 91, the bases 3, 4, 9, 10, 12, 16, 17, 22, 23, 25, 29, 30, 36, 38, 40, 43, 48, 51, 53, 55, 61, 62, 66, 68, 69, 74, 75, 79, 81, 82, 87, 88 also exist, but neither of them lies between a multiple of 7 and a multiple of 13.(End)

Looking at odd squarefree semiprimes less than 10000, it appears that the number of bases is always of the form 2(2k^2-1), which is A060626 and twice A056220. Using the formula in A063994, the number of bases for pq (including bases 1 and pq-1) is gcd(p-1,pq-1) * gcd(q-1,pq-1).

LINKS

Table of n, a(n) for n=1..93.

FORMULA

a(n) = A063994(A046388(n)) - 2.

EXAMPLE

For A046388(1) = 15, the bases b in the range [2,13] are 4 and 11. So a(1) = 2.

CROSSREFS

Cf. A046388, A063994 (number of bases b for which b^(n-1) = 1 (mod n))

Sequence in context: A123627 A171818 A248777 * A001991 A305706 A147793

Adjacent sequences:  A175098 A175099 A175100 * A175102 A175103 A175104

KEYWORD

nonn

AUTHOR

T. D. Noe, Dec 02 2010

STATUS

approved

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Last modified January 18 20:57 EST 2019. Contains 319282 sequences. (Running on oeis4.)